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Properties

Label 2-48-3.2-c24-0-25
Degree $2$
Conductor $48$
Sign $-0.598 - 0.801i$
Analytic cond. $175.184$
Root an. cond. $13.2357$
Motivic weight $24$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.17e5 + 4.25e5i)3-s + 6.76e7i·5-s + 2.41e10·7-s + (−8.02e10 + 2.70e11i)9-s + 4.00e12i·11-s + 2.67e13·13-s + (−2.88e13 + 2.15e13i)15-s + 6.60e14i·17-s + 3.05e14·19-s + (7.68e15 + 1.02e16i)21-s − 7.66e14i·23-s + 5.50e16·25-s + (−1.40e17 + 5.19e16i)27-s + 4.00e17i·29-s − 1.08e18·31-s + ⋯
L(s)  = 1  + (0.598 + 0.801i)3-s + 0.277i·5-s + 1.74·7-s + (−0.284 + 0.958i)9-s + 1.27i·11-s + 1.14·13-s + (−0.222 + 0.165i)15-s + 1.13i·17-s + 0.137·19-s + (1.04 + 1.39i)21-s − 0.0349i·23-s + 0.923·25-s + (−0.938 + 0.345i)27-s + 1.13i·29-s − 1.37·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.598 - 0.801i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (-0.598 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.598 - 0.801i$
Analytic conductor: \(175.184\)
Root analytic conductor: \(13.2357\)
Motivic weight: \(24\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :12),\ -0.598 - 0.801i)\)

Particular Values

\(L(\frac{25}{2})\) \(\approx\) \(4.086239390\)
\(L(\frac12)\) \(\approx\) \(4.086239390\)
\(L(13)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-3.17e5 - 4.25e5i)T \)
good5 \( 1 - 6.76e7iT - 5.96e16T^{2} \)
7 \( 1 - 2.41e10T + 1.91e20T^{2} \)
11 \( 1 - 4.00e12iT - 9.84e24T^{2} \)
13 \( 1 - 2.67e13T + 5.42e26T^{2} \)
17 \( 1 - 6.60e14iT - 3.39e29T^{2} \)
19 \( 1 - 3.05e14T + 4.89e30T^{2} \)
23 \( 1 + 7.66e14iT - 4.80e32T^{2} \)
29 \( 1 - 4.00e17iT - 1.25e35T^{2} \)
31 \( 1 + 1.08e18T + 6.20e35T^{2} \)
37 \( 1 - 2.67e18T + 4.33e37T^{2} \)
41 \( 1 + 3.10e19iT - 5.09e38T^{2} \)
43 \( 1 - 2.43e19T + 1.59e39T^{2} \)
47 \( 1 + 3.79e19iT - 1.35e40T^{2} \)
53 \( 1 - 8.82e19iT - 2.41e41T^{2} \)
59 \( 1 - 1.88e21iT - 3.16e42T^{2} \)
61 \( 1 + 1.92e21T + 7.04e42T^{2} \)
67 \( 1 - 7.96e21T + 6.69e43T^{2} \)
71 \( 1 + 2.04e22iT - 2.69e44T^{2} \)
73 \( 1 - 3.28e22T + 5.24e44T^{2} \)
79 \( 1 + 5.13e21T + 3.49e45T^{2} \)
83 \( 1 - 1.14e21iT - 1.14e46T^{2} \)
89 \( 1 - 3.69e23iT - 6.10e46T^{2} \)
97 \( 1 - 1.10e24T + 4.81e47T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.99266492197095490363190556533, −10.61138148705286915735662786541, −9.088987243853229207973666190886, −8.263474306535030439495525033752, −7.25415382873012929216266197920, −5.48460351265355496670460887581, −4.52950037008140282987172968625, −3.64915185698601655842630766009, −2.13875778409714647913744707543, −1.42174331039766793470849321307, 0.72262708069637397947083050219, 1.28554146366938594605158066695, 2.45421723702775386213338853991, 3.68024868691063704852278473240, 5.05207262023319142041213748735, 6.19544352979454466572084894618, 7.62984866162361658514738826901, 8.335987116916472039082571441624, 9.130119760695007516106925578209, 11.10647685959855519604696322231

Graph of the $Z$-function along the critical line